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Forschungsinstitut zur Zukunft der ArbeitInstitute for the Study of Labor

Real Wages and the Business Cycle in Germany

IZA DP No. 5199

September 2010

Martyna MarczakThomas Beissinger

Real Wages and the

Business Cycle in Germany

Martyna Marczak University of Hohenheim

Thomas Beissinger

University of Hohenheim and IZA

Discussion Paper No. 5199 September 2010

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IZA Discussion Paper No. 5199 September 2010

ABSTRACT

Real Wages and the Business Cycle in Germany* This paper establishes stylized facts about the cyclicality of real consumer wages and real producer wages in Germany. As detrending methods we apply the deterministic trend model, the Beveridge-Nelson decomposition, the Hodrick-Prescott filter, the Baxter-King filter and the structural time series model. The detrended data are analyzed both in the time domain and in the frequency domain. The great advantage of an analysis in the frequency domain is that it allows to assess the relative importance of particular frequencies for the behavior of real wages. In the time domain we find that both real wages display a procyclical pattern and lag behind the business cycle. In the frequency domain the consumer real wage lags behind the business cycle and shows an anticyclical behavior for shorter time periods, whereas for longer time spans a procyclical behavior can be observed. However, for the producer real wage the results in the frequency domain remain inconclusive. JEL Classification: E32, C22, C32, J30 Keywords: real wages, business cycle, frequency domain, time domain, Germany,

trend-cycle decomposition, structural time series model, phase angle Corresponding author: Martyna Marczak University of Hohenheim Department of Economics Schloss, Museumsfluegel D-70593 Stuttgart Germany E-mail: marczak@uni-hohenheim.de

* We thank Anita C. Bott, Jacques J.F. Commandeur, Irma Hindrayanto, Kurt Jetter, the participants of the WIEM 2010 conference in Warsaw and the participants of the annual congress of the German Economic Association 2010 in Kiel for helpful comments.

1 Introduction

At least since Keynes (1936) claimed in his General Theory that an increase in em-

ployment can only occur with a simultaneous decline in real wages, macroeconomists

are debating about whether real wages are anticyclical, procyclical or do not exhibit

any systematic relationship with the business cycle. A clarification of this issue

could shed some light on the main sources of macroeconomic shocks and thereby

be of some use in judgements about the empirical relevance of conflicting macroe-

conomic theories. A clearer empirical picture about the adjustment of real wages

over the business cycle also helps in identifying the sources and features of wages

and labor cost dynamics and therefore is of great relevance for monetary policy.

This paper contributes to the literature on the adjustment of aggregate real

wages over the business cycle in several ways. First, we analyze the comovements

between real wages and the cycle not only in the time domain, but also in the

frequency domain. So far, most studies have focussed on the time domain approach

and described the comovements between variables by traditional cross–correlations

measures. However, as has been pointed out by Hart et al. (2009), in the time

domain the observed cyclical behavior of the real wage hides a range of economic

influences that give rise to cycles of different length and strength, thereby producing

a distorted picture of real wage cyclicality. The great advantage of an analysis in

the frequency domain is that it allows to assess the relative importance of particular

frequencies for the behavior of real wages.

Second, it is analyzed whether the empirical results are robust to the method

used to extract the cycle from the data. More specifically, we apply the deterministic

trend model, the Beveridge–Nelson decomposition, the Hodrick–Prescott filter, the

Baxter–King filter and the structural time series model of Harvey (1989) to the time

series of aggregate real wages and gross domestic product. Since it is well known

from the literature that the results may also be influenced by the price deflator used

1

to compute real wages, we take this into account by considering both producer real

wages and consumer real wages.

Third, we analyze the real wage behavior for the economy as a whole whereas

many studies only consider real wages in the manufacturing sector, as for example

in the recent study of the wage dynamics network of the ECB on real wage behavior

in the OECD, see Messina et al. (2009).1 Because of the much larger shares of the

non-manufacturing sector in total output and employment, empirical results for the

economy as a whole are certainly preferable.

Forth, whereas the question of the cyclicality of real wages in the US has been

analyzed in a host of studies (see the surveys of Abraham and Haltiwanger, 1995,

and Brandolini, 1995), surprisingly little systematic empirical evidence exists for

Germany. This paper tries to fill this gap and provides a detailed picture of the

wage dynamics in an economy in which labor unions (still) affect the majority of

employment contracts.

The remainder of the paper is organized as follows. In Section 2 we describe

our data and then analyze the stochastic properties of the time series. In Section 3

different trend–cycle decompositions are applied to consumer real wages, producer

real wages and real GDP. In Section 4 we analyze the comovements between the

particular real GDP cycle and the corresponding real wage cycles in the time and

frequency domain. Section 5 summarizes and concludes.

2 Data and stochastic properties of the series

We use quarterly data for real GDP, consumer real wages and producer real wages

in Germany from 1970.Q1 to 2009.Q1 (157 observations). All series that served to

generate the project data were seasonally adjusted with the Census-X12-ARIMA

1In Messina et al. (2009) also time domain and frequency domain methods are used.

2

procedure. The data prior to 1991.Q1 refers to West Germany and has been linked

to the data of unified Germany using annual averages for 1991. The data selection

is described in more detail in Appendix A. All generated data are represented in

natural logarithms.

Before we undertake the trend-cycle decompositions we study the stochastic

properties of the data. We test for unit roots in real GDP and both real wage series

applying the augmented Dickey-Fuller (ADF) test and the Phillips-Perron test. In

both tests the alternative hypothesis is based on the assumption that the particular

series follows a trend–stationary process since all series exhibit a clear upward course.

The lag length for the ADF test is determined by the Akaike information criterion

(AIC) and Schwartz information criterion (SIC) and its accuracy is then verified

with the Ljung-Box test and the Breusch-Godfrey test. The results of both unit

root tests show that for all series the unit root hypothesis cannot be rejected using

conventional significance levels. Therefore, the underlying stochastic processes are

not covariance stationary. We then apply both unit root tests to the first differences

of the series. Since the null hypothesis can now be rejected, we conclude that all

series are generated by I(1) processes.

3 Identification of the cyclical component

The general framework for the decomposition of each time series into trend and

cycle is provided by the following model:

yt = ygt + yc

t + εt, t = 1, 2, ..., T (1)

where t is a time index and yt represents the natural logarithm of the series under

consideration, i.e. real GDP, consumer real wages or producer real wages. The series

yt is decomposed into trend ygt , cycle yc

t and (possibly) an irregular component εt.

The latter is only relevant in the structural time series model (STSM), whereas it

3

is neglected in the other decomposition methods applied in this paper, namely the

linear trend model with broken trend (LBT), the Beveridge–Nelson decomposition

(BN), the Hodrick–Prescott filter (HP) and the Baxter–King filter (BK). The latter

methods assume the variance of εt to be zero, thereby attributing any disturbance

left in the data after the removal of the trend to the cyclical component.

As a first decomposition method we consider the linear trend model. To check

whether the time series under consideration is subject to structural breaks we apply

the Quandt-Andrews test.2 For all time series the test clearly rejects the hypothesis

of no structural break. The estimated break point is 2002.Q4 for real GDP and

2003.Q1 for both real wages. Based on this result we estimate the following model:

yt = α0 + α1t + β1St,k + νt,

St,k =

t− k, if t > k

0, if t ≤ k,

(2)

where νt is generated by a covariance stationary process which is uncorrelated with

{yt} and St,k reflects the change in the slope of the trend starting with period k.

The results of the OLS estimation of model (2) are reported in Table 1. According

to these findings real GDP grows with a (quarterly) rate of 0.57% until 2002.Q4.

From 2002.Q4 on its growth slows down by 0.47 percentage points. The growth rate

of the consumer real wage equals 0.35% before and -0.33% after the break point.

The growth rate of the producer real wage exceeds that of the consumer real wage

by 0.17 percentage points over the first period. However, after 2003.Q1 it falls more

steeply than in the case of the consumer real wage. The deviations from the growth

path, i.e. the residuals of the estimated model, represent the cyclical component,

hence yct = νt. The examination of the residuals with correlograms and the Ljung-

2See Andrews (1993). This test overcomes the shortcomings of the commonly used Chow test

in that it does not require any previous knowledge about the occurrence of a possible break point.

We choose standard 15% as “trimming” level for this test.

4

Table 1: Estimation of segmented linear trend models for real

GDP and real wages

regressorGDP consumer wage producer wage

coefficients a)

t0.0057 0.0035 0.0052

(104.21) (59.19) (46.87)

Sk,t

−0.0047 −0.0068 −0.0112

(−10.49) (−13.4) (−11.77)

constant5.544 2.385 2.173

(1278.237) (510.194) (246.321)

a) t-values in parentheses. Number of observations: 157. Break point

k = 132 for real GDP and k = 133 for real wages.

Box test indicate that the obtained cycles of real GDP and real wages follow a

persistent AR(1) process. This is confirmed by the results of the ADF test applied

to each of these cycles. The hypothesis of a unit root cannot be rejected at the 5%

significance level in each case. Since the LBT cycles do not satisfy the stationarity

condition which is needed for the comovement analysis, we exclude the LBT cycles

from further analysis.

As has been shown in Section 2, both real GDP and real wages are difference–

stationary processes. For this case, a suitable decomposition method has been sug-

gested by Beveridge and Nelson (1981). The BN decomposition assumes a I(1)

process for the examined series and regards the trend as a prediction of future val-

ues of the series. The decomposition leads to a trend component which is a random

walk with drift and to a covariance stationary cyclical component which are cor-

related with each other. In this respect, the BN decomposition differs from the

5

LBT model with its strong assumption of zero correlation between trend and cycle.

However, the BN decomposition also bears some problems. For example, it requires

an ARMA specification for the examined series but since distinct ARMA models

can suit the data, different forecasts can result from these models. That in turn

implies different trends and cycles. Furthermore, the a priori assumption about the

trend being a random walk is somewhat controversial. Another problematic issue

concerns the variance of the trend that could even exceed that of the series.

The procedure determining the trend and cycle requires truncation of infinite

sums and is associated with heavy computational burden. In order to reduce these

costs, different resolution methods have been proposed in the literature.3 In this

paper, we take the approach of Newbold (1990) which is based on the ARIMA(p, 1, q)

representation of the series.4 The BN cycle can be described as:

yct =

q∑j=1

[zt(j)− µ] + (1− φ1 − ...− φp)−1

p∑j=1

p∑i=j

φi[zt(q − j + 1)− µ], (3)

where zt(k) denotes the k-periods ahead forecast of z = ∆y made in period t. φj is

the AR coefficient at lag j and µ is the mean of the process {zt}. To isolate the cycle

using (3) we have to identify the best ARMA specification for the first differences

of real GDP and real wages. In doing so, we rely on the information criteria (AIC

and SIC) beginning with an AR(4) model for each of the series in first differences.

Since ARMA modeling technique aims at a parsimonious representation we reduce

the number of AR terms and then optionally add some MA terms and compare all

models with regard to the values of AIC and SIC. The initially considered AR(4)

specification turns out to be the most suitable one. Next, we examine the residuals

3See, for example, Cuddington and Winters (1987), Miller (1988) and the more recent work of

Morley (2002).4According to Wold’s theorem, each covariance stationary process has a MA(∞) representation

which is also consistent with an ARMA(p, q) representation. Therefore, each I(1) process in first

differences has an ARMA(p, q) representation.

6

from this model with the Ljung-Box test and the Breusch-Godfrey test. We find

no evidence for serial correlation of the residuals in the case of real GDP as well as

the real producer wage in first differences, respectively. Hence, for these series we

choose an AR(4) model. As for the first differences of the real consumer wage, the

residual autocorrelation vanishes after including an additional lag, so we finally end

up with an AR(5) specification. Inserting the forecasts based on the selected models

in (3) yields the cyclical components of real GDP and real wages.

As the next trend-cycle decompositions we use linear filters, the HP filter and

the BK filter, which have proven popular in macroeconomic applications.5 A great

advantage of these methods can be seen in the fact that they are able to render the

data stationary. They also avoid modeling of the series in contrast to, e.g. , the BN

decomposition. However, the results of both filters are not without problems if they

are applied to series which are generated by nonstationary processes. It has been

shown in the literature that in this case the HP filter induces spurious cycles.6 This

is due to the fact that the frequency components of the resulting series have business

cycle periodicity even though there are no important transitory fluctuations in the

original data. As regarding the critique of the BK filter application for nonstationary

series, Murray (2003) demonstrates that the first difference of an integrated trend

enters the filtered series. As a result, the spectral properties of the filtered series

depend on the trend in the unfiltered series. Because of the nonstationarity of

the analyzed series the cycles obtained with the HP and the BK filter should be

interpreted with some caution.

Finally, we consider structural time series models, which are defined in terms

of unobserved components that have a direct economic interpretation.7 The initial

5As suggested by Hodrick and Prescott (1980), for the HP filter we use the value 1600 for the

smoothing parameter.6See, for example, Cogley and Nason (1995) and Harvey and Jaeger (1993).7See Harvey (1989, pp. 44–49).

7

specification of the model structure is left to the researcher. Within this framework,

the data decide on the characteristics of the particular component. In contrast to the

ad hoc filtering approaches, such as the HP and the BK filter, structural time series

models rely on the stochastic properties of the data. Moreover, as opposed to ARMA

modeling they do not aim at a parsimonious specification. It is quite probable

that a parsimonious ARMA model identified by means of standard techniques (e.g.

correlograms) does not exhibit properties expected from the examined series. For

instance, it could reject cyclical behavior of a series even though such a behavior

does really exist. Unfortunately, finding a “correct” model specification inevitably

also remains a problem in the case of structural time models.

In this paper, we adopt the model as in eq. (1) and assume that the irregular

component εt is normally, independent and identically distributed with variance σ2ε :

εt ∼ NID(0, σ2ε) (4)

The stochastic trend component ygt can be formulated as follows:8

ygt+1 = yg

t + βt + ηt, ηt ∼ NID(0, σ2η)

∆mβt+1 = (1− L)mβt+1 + ζt, ζt ∼ NID(0, σ2ζ )

(5)

The variable βt is the slope of the trend and the scalar m (m = 1, 2, 3, ...) is the

order of the slope. If the slope follows an I(m) process then the trend is I(m + 1).

In case of m = 1 the trend is called local linear trend. Imposing restrictions on the

variances σ2η and σ2

ζ leads to various trend forms. With m = 1 and σ2η = σ2

ζ = 0 one

obtains a deterministic linear trend. The assumption σ2ζ = 0 together with m = 1

implies that the trend is a random walk, whereas σ2η = 0 along with m = 1 results in

a relatively smooth I(2) trend component. If one needs to model an even smoother

trend component the trend is supposed to be of higher order (m > 1). The cycle yct

8See Koopman et al. (2009, S. 55-56).

8

is defined as:9

yc

t+1

yc∗t+1

= ρ

cos(ω) sin(ω)

− sin(ω) cos(ω)

yc

t

yc∗t

+

χt

χ∗t

,

χt

χ∗t

∼ NID(0, σ2

χI2),

(6)

where yc∗t is an auxiliary variable, ω denotes the frequency (0 ≤ ω ≤ π) and ρ is

the damping factor (0 ≤ ρ ≤ 1). The period p of the cycle is therefore p = 2π/ω.

If ω = 0 or ω = π, the VAR(1) process in (6) collapses into an AR(1) process. The

variance σ2χ is given as σ2

χ = σ2c (1 − ρ2), where σ2

c is the variance of the cycle so

that with ρ = 1 the cycle is reduced to a deterministic and covariance stationary

process. For all three series we start with the general formulation of the model with

no variance restriction (model 1), but we constrain the trend specification to the

local linear trend (m = 1). The whole model is estimated by maximum likelihood

with the Kalman filter. The Kalman smoothing provides the estimates of the trend

and cycle component. The estimated model parameters, called hyperparameters,

that refer to real GDP are reported in Table 2. A high value of σ2η relative to σ2

χ

indicates an erratic trend component and a damped cycle component. Since this

result seems rather implausible, in the next step we apply the restriction σ2η = 0

ensuring bigger deviations of the cycle than in the general model (see Table 2, model

2). The cycle extracted from the restricted model does coincide better with the

German history of booms and recessions. The results of a likelihood ratio (LR) test

also confirm the validity of the variance restriction. For the consumer real wage, the

initial model leads to a deterministic cyclical component so in this case we reject the

general model, too (see Table 3). We enforce the cyclical component to be stochastic

9See Koopman et al. (2009, p. 63), Koopman et al. (2008, p. 23) und Harvey and Streibel

(1998). Clark (1989) suggests to describe the cycle as an AR(2) process. Harvey and Trimbur

(2003) generalize the trigonometric version in (6) to cycles of higher order.

9

Table 2: Estimation of the general and restricted trend-cycle model for

real GDP

modelhyperparametera)

R2D

b)

σ2ε σ2

η σ2ζ σ2

χ ρ ω

1) no restrictions 0,865 74,98 0,215 11,734 0,976 0,234 0,0545

2) σ2η = 0 5,032 – 0,449 67,683 0,929 0,194 0,0469

a) The estimated variances have been multiplied by 106.b) The coefficient of determination R2

D is based on the first differences of the ob-

served series.

Table 3: Estimation of three trend-cycle models for the consumer real wage

modelhyperparametera)

R2D

b)

σ2ε σ2

η σ2ζ σ2

χ ρ ω

1) no restrictions, m = 1 24,187 22,883 1,862 0 1 0,499 0,0964

2) σ2η = 0, m = 1 33,166 – 3,298 1,102 0,966 0,468 0,0697

3) σ2η = 0, m = 2 22,482 – 0,004 26,537 0,957 0,153 0,0498

a) The estimated variances have been multiplied by 106

b) The coefficient of determination R2D is based on the first differences of the observed series

by assuming σ2η = 0. However, the irregular term becomes the most important

component in explaining the consumer real wage variation and the cycle has a high

frequency (see Table 3, model 2). These problems are eliminated if we allow for

a smoother trend by setting m = 2 (see Table 3, model 3). We proceed similarly

with the model identification for the producer real wage, hence we choose the trend-

cycle model with σ2η = 0 and m = 2. The estimation results are summarized

10

in Table 4. Following e.g. Harvey and Koopman (1992) and Commandeur and

Table 4: Estimation of three trend-cycle models for the producer real wage

modelhyperparametera)

R2D

b)

σ2ε σ2

η σ2ζ σ2

χ ρ ω

1) no restrictions, m = 1 14,894 61,212 9,687 0 1 0,542 0,119

2) σ2η = 0, m = 1 30,67 – 12,293 10,776 0,928 0,505 0,0901

3) σ2η = 0, m = 2 10,39 – 0,009 73,623 0,947 0,203 0,0914

a) The estimated variances have been multiplied by 106

b) The coefficient of determination R2D is based on the first differences of the observed series

Koopman (2007), we then check all selected models with the following diagnostic

tests: the Ljung-Box autocorrelation test, the Goldfeld-Quandt heteroscedasticity

test and the Bowman-Shenton normality test. In all cases, we cannot reject the

hypothesis of no autocorrelation at the 5% significance level. The heteroscedasticity

test finds evidence against the homoscedasticity assumption only for the consumer

real wage, whereas the Bowman-Shenton test indicates violation of the normality

assumption for all series. Nevertheless, since the main concern of time series analysis

is the autocorrelation problem we can conclude that these models provide a satisfying

specification of the data generating process.

In Figure 1, we depict the cyclical component for real GDP for the different

detrending methods outlined above. One can easily recognize the periods of booms

and recessions that Germany has experienced since 1970. The first recessions occur

as a result of the first and second oil crisis 1974–75 and 1980–82.10 After a relatively

weak recovery in the second half of the 1980s one can observe a clear boom phase

10We do not interpret the large negative values at the beginning of the sample in the case of the

LBT and HP cycles since they could be caused by problematic behavior of these methods at the

bounds.

11

in the early 1990s that is due to German reunification. The next turning point is

reached in 1993 with the beginning of a recession phase initiated by the restrictive

monetary policy of the Deutsche Bundesbank. In the late 1990s one observes a

recovery that may have been caused by the IT boom followed by a recessionary

phase 2001–2005. Afterwards, the economy expands again. This positive course

ends in 2008 because of the world economic and financial crisis leading to a severe

downturn. The various cyclical components of the consumer real wage and the

Figure 1: Cycles of real GDP

LBT BK

HP

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.05

0.00

0.05LBT BK

HP

BN STSM

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.050

−0.025

0.000

0.025

0.050BN STSM

producer real wage are compared in Figure 2 and Figure 3, respectively.

For all series, the LBT cycles exhibit the most striking peaks and troughs. It is

12

Figure 2: Cycles of the consumer real wage

LBT BK

HP

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.05

0.00

0.05LBT BK

HP

BN STSM

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.05

0.00

0.05

BN STSM

apparent that the BN cycles of both real wages are shifted relative to the STSM,

HP and BK cycles. Moreover, the STSM cycles of both real wages are almost in line

with the HP cycles. This can be explained by the fact that the structural time series

model with a trend of higher order can be associated with a Butterworth filter.11

11Gomez (2001) shows that the HP filter is a Butterworth filter.

13

Figure 3: Cycles of the producer real wage

LBT BK

HP

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.10

−0.05

0.00

0.05

0.10LBT BK

HP

BN STSM

1970 1975 1980 1985 1990 1995 2000 2005 2010

−0.1

0.0

0.1

BN STSM

4 Comovements of real GDP and real wages

4.1 Time Domain

The analysis of comovements in the time domain between real wage cycles and real

GDP cycles as a reference for the business cycle is a natural approach to detect the

cyclical behavior of real wages. The measure of comovements we adopt here are the

sample cross–correlations between the cycle of each of the real wage series and the

real GDP cycle. We consider not only the contemporaneous relationship but also

analyze whether real wages react with delay or run ahead of cyclical movements in

14

real GDP. We find it a bit misleading that in the literature wages are often classified

as pro– or countercyclical by solely focusing on the contemporaneous correlation.12

Using quarterly data, one can not seriously expect that the main adjustment of real

wages to cyclical changes in GDP will take place in the same period. Therefore, we

classify the considered real wage as procyclical (countercyclical) if the estimated cor-

relation coefficients are positive (negative) taking into account the lead–lag structure

of the examined series. If the estimated correlation coefficients are close to zero the

particular real wage is defined to be acyclical. If the largest sample cross–correlation

occurs at any lead (lag) relative to the GDP cycle we say that the particular real

wage is lagging (leading) the cycle.

Table 5: Contemporaneous and largest sample cross–correlations between the

real GDP cycle and the particular real wage cycle by various decom-

position methods

correlation of methods

GDP with BN HP BK STSM

consumer real wage0,1169 0,0124 0,1438 −0, 1677∗

0, 4879∗(+6) 0, 4572∗(+6) 0, 6346∗(+5) 0, 4099∗(+11)

producer real wage0,0279 −0, 0423 0,0314 −0, 0362

0, 2718∗(+6) 0, 2381∗(+7) 0, 315∗(+7) 0, 2163∗(+10)

Notes: “ * ” indicates statistical significance at the 5% level

The findings are summarized in Table 5. Each cell contains in the first row the

contemporaneous sample cross–correlation between the cycle of the real wage series

and the corresponding real GDP cycle. The value below is the maximum sample

12This focus of the literature on the contemporaneous correlation has also been criticized by

Messina et al. (2009).

15

cross–correlation at the kth lead or lag of the real wage cycle relative to the real GDP

cycle, where k ∈ {−12,−11, ..., 0, ..., 11, 12}. The number in brackets along with the

“+” or “−” sign specifies at which lead or lag of the real wage cycle this maximum

cross–correlation occurs.13 We first consider the results for the consumer real wage.

Except for the STSM cycle, the estimates of the contemporaneous cross–correlation

are positive but statistically insignificant at the 5% level. The low practical signifi-

cance is most apparent in the case of the HP cycle. Considering the leads of the real

wage cycles, we find that for all cycles except for the STSM cycles, the relationship

with the corresponding real GDP cycles is still positive but now becomes significant.

The sample cross–correlations reach their maximum values at the 6th lead (BN and

HP cycles) or 5th lead (BK cycles). In the case of the STSM cycles, there is first a

significant negative sample cross–correlation until the 3rd lead. From the 6th lead,

it takes high positive values that are statistically significant. We find the greatest

cross–correlation at the 11th lead. Examination of the lags of the real wage cycles

reveals that almost all sample cross–correlations are statistically insignificant in the

case of the HP, BK and BN cycles. The significant ones are small compared to

the significant sample cross–correlations at the leads.14 To sum up, the consumer

real wage displays a procyclical pattern and lags the business cycle. The strongest

reaction to the actual economic situation can be observed between the 5th and the

11th quarter.

The behavior of the producer real wage differs somewhat from that of the con-

sumer real wage. All estimated contemporaneous cross–correlations are statistically

insignificant at the 5% level. Furthermore, although there is a similar cyclical pattern

13For clarity reasons, we do not present detailled figures of the lead–lag structure and instead

describe some results verbally.14In the case of the STSM cycles, the significant negative sample cross–correlations emerge at

the first 3 lags. In contrast, the BN cycles are characterized by positive cross–correlations which,

though, are insignificant.

16

as in the case of the consumer real wage, the sample cross–correlations at the leads

of the real wage are not as high. In Table 5 this is evident from the differences in the

maximum cross–correlations between both wages. The sample cross–correlations at

the lags of the producer real wage, with the exception of some lags in the case of the

BN cycles, are statistically insignificant. The analysis leads to the conclusion that

the producer real wage behaves procyclically and lags the business cycle. The main

reaction to the actual economic situation occurs after 6 (BN cycle) to 10 (STSM

cycle) quarters.

4.2 Frequency domain

The above analysis of the comovements between real wages and the cycle in the

time domain might give the impression that the cyclicality of real wages has been

sufficiently characterized. However, the observed behavior of real wages in the time

domain results from the countervailing or/and reinforcing influences of cycles of

different length. As a consequence, if we want to learn something about the behavior

of real wages over the business cycle, we could be misled by looking at the time

domain results alone. In this section, we resort to some spectral analysis concepts

that enable us to assess the relative importance of cycles of different length and

strength and therefore provide a comprehensive picture on the cyclical behavior of

real wages.

We will first give a short introduction to these concepts. The central one is the

spectral representation of a covariance stationary process, also called the Cramer

representation, as a frequency domain counterpart to the Wold representation of

such a process. According to the spectral representation a time series Yt, which

is a single realization of a zero–mean covariance stationary process yt, is regarded

as comprising various superimposed cosine and sine waves each having different

frequency and amplitude. If such a stochastic process yt is discrete and real–valued,

17

it can be described by:15

yt =

∫ π

0

α(ω) cos(ωt)dω +

∫ π

0

δ(ω) sin(ωt)dω, (7)

where α(ω) and δ(ω) are orthogonal complex–valued stochastic processes with zero

mean and equal variances. The variable ω denotes the (angular) frequency. The

coefficients α(ω) and δ(ω) give rise to the stochastic nature of the process in (7), in

that |α(ω)| and |δ(ω)| are random amplitudes, whereas arg{α(ω)} and arg{δ(ω)} de-

scribe random phases of the particular cosine and sine wave. Each wave contributes

to the explanation of the overall variance (power) of this process. This information

is given by the real–valued function s(ω), the so–called spectral density function or,

in short, spectrum:

s(ω) =1

2π

∞∑j=−∞

γj e−iωj

=1

2π

∞∑j=−∞

γj cos(ωj)

=1

2π

[γ0 + 2

∞∑j=1

γj cos(ωj)],

(8)

where γj is the jth autocovariance of the process and i is the imaginary number.

The area under the graph of s(ω) for ω ∈ [−π, π] describes the total variance of the

process.

Since we are primarily interested in the interactions between time series, we now

turn to the multivariate case and consider two series Ykt and Ylt (k, l = 1, 2, .., n, k 6=l). The frequency by frequency relationship between the underlying processes ykt

and ylt can be measured by the cross–spectrum skl(ω):

skl(ω) =1

2π

∞∑j=−∞

γjkl e−iωj

=1

2π

∞∑j=−∞

γjkl[cos(ωj)− i sin(ωj)],

(9)

15See, for example, DeJong and Dave (2007, pp. 41–42) and Priestley (1981, pp. 251–252).

18

where γjkl is the jth cross–covariance of the two processes defined as

γjkl = E[(ykt − µk)(yl,t−j − µl)] (10)

The cross–spectrum, which is a complex–valued function of ω, can be decomposed

into the real part ckl(ω) called cospectrum and the imaginary part qkl(ω) called

quadrature spectrum. In analogy to the spectrum of an individual process, the

area under the cross–spectrum in the range [−π, π] gives the overall covariance of

the two processes. Additionally, as the quadrature spectrum integrates to zero in

this interval, the area under the cross–spectrum is equal to the area under the

cospectrum. According to this, the cospectrum at frequency ω can be interpreted as

the marginal contribution of the components with frequency ω + dω to the overall

covariance between the processes. The quadrature spectrum at this frequency can

serve as an indicator for the out–of–phase covariance since it measures the portion

of the covariance between two processes shifted relative to one another by π/2 which

is attributable to the waves with this frequency. The information contained in both

the cospectrum and the quadrature spectrum is useful in establishing the lead–lag

relationship between two processes. For this purpose, the inferences from both parts

of the cross–spectrum at any frequency can be combined into one quantity, the so–

called phase angle, denoted by θ(ω):

θ(ω) = arctan

[qkl(ω)

ckl(ω)

](11)

Because of the properties of arctangent, the phase angle θ(ω) is a multivalued func-

tion, but it is common to limit its values to the interval (−π, π). The unique value

in (−π, π) and therewith the sign of the phase angle can, however, only be deter-

mined by the signs of the cospectrum and the quadrature spectrum. If θ(ω) takes

on positive values, we say that the component of ykt with frequency ω leads the

corresponding component of ylt. The opposite case is implied by θ(ω) < 0. Both

components are in phase if θ(ω) equals zero. Based on the values of the phase angle

19

we can also make statements about the correlation between ykt and ylt. If the values

of the phase angle range between [−π/2, π/2], we say that ykt and ylt are positively

correlated (procyclical behavior), whereas the values of θ(ω) in the interval [π/2, π]

or [−π/2,−π] indicate a negative relationship (countercyclical behavior) between

them.

In this paper, we focus on the nonparametric approach to the estimation of

spectra and cross–spectra.16 For that purpose, we have to choose a suitable spectral

window and the truncation point of the window. Since, as pointed out by Jenkins

and Watts (1968, p. 280), the spectral estimates are barely affected by the functional

form of the window, we use the Bartlett window and concentrate on finding an

appropriate truncation point. We allow the window lag size to be 20 resting upon

the technique of window closing which enables a researcher to make her choice in

the process of learning about the shape of the spectrum instead of relying on any

rules of thumb.17 The estimated spectra of the HP, BK, BN and STSM cycles of real

GDP, consumer real wages and producer real wages, and the cross–spectra between

all GDP cycles and the real wage cycles are shown in Appendix B.

In the following, we focus on the interpretation of the estimated phase angle.

In Figures 4 and 5, the point estimates of the phase lead of the real GDP cycle

over the corresponding consumer and producer real wage cycle, respectively, for all

decomposition methods along with the respective confidence bounds are drawn.18

The frequency range presented here covers all business cycle periodicities, i.e. periods

between 1.5 (frequency of about 1.0) and 8 years (frequency of about 0.2).19 The

relationship between frequency ω and period p is given by the formula: p = 2π/ω. It

16We refer those readers who are not familiar with univariate and multivariate spectral estimation

to, e.g. , Koopmans (1974, Ch. 8) and Priestley (1981, Ch. 6 and 9.5).17The method of window closing is described in Jenkins and Watts (1968, pp. 280–282).18We construct the 90% confidence intervals as described in Koopmans (1974, pp. 285–287).19Following the seminal paper of Burns and Mitchell (1946) this is the commonly used range for

the business cycle length.

20

should be noticed that the vertical axis representing the values of the phase angle is

divided into four regions.20 If the confidence interval covers one of two upper regions

we say that the real GDP cycle significantly leads the real wage cycle. The opposite

holds true if the confidence interval lies in one of the two lower regions. A significant

procyclical behavior of the real wage cycle is indicated by the confidence interval in

the two regions around 0. If, on the other hand, the confidence interval covers the

top or the bottom region we conclude that the real wage behaves countercyclically.

If the confidence interval covers at least three regions, we interpret it as being a “no

information confidence interval”.

In Figure 4, it is apparent that for the consumer real wage the point estimates

of the phase angle display a similar pattern in the case of the HP, BK and BN

cycles. At all frequencies, the estimated phase angle takes on positive values which

suggests a lagging behavior of cycles of the real wage characterized by business

cycle frequencies with respect to the corresponding cycles of real GDP. However,

statistical significance of such a behavior pertains rather to lower business cycle

frequencies. We also observe that for these three decomposition approaches the lower

frequencies (up to about 0.35) are associated with estimates of the phase angle in

the interval [0, π/2]. The significant ones are confined to the frequencies up to 0.25

thereby indicating significant procyclical pattern of the consumer real wage at these

frequencies. In contrast, shorter cycles of the real wage are negatively correlated

with the particular real GDP cycle as shown by the estimated phase angle values

lying above π/2. For the STSM cycles we obtain positive point estimates at lower

frequencies as well. However, we cannot make any statement about the statistical

significance of any estimated phase angle in the whole frequency range. Taking all

findings into account we can conclude that, in general, longer consumer real wage

20The results for each frequency are illustrated on a linear scale which can be obtained through

“straightening” a circular scale connected by the points representing angles π and −π.

21

Figure 4: Phase angle: real GDP and consumer real wage cycles

0.0 0.2 0.4 0.6 0.8 1.0

HP cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

BK cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

BN cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

STSM cyclesπ

−π

π/2

−π/2

0

Notes: The horizontal axis represents (angular) frequency ω.

cycles seem to exhibit a procyclical and lagging behavior, whereas the shorter ones

evolve countercyclically and also react with delay to the actual economic situation.

As for the producer real wage, the estimation results presented in Figure 5 look

almost identical to the ones for the consumer real wage if we consider the estimated

values of the phase angle at lower frequencies. Also, the values in the interval

[−π,−π/2] (HP, BK and BN cycles) at the frequencies above 0.4 (below 4 years)

could serve as an indicator for a countercyclical behavior of the producer real wage.

Despite this similarity to the correlation scheme of the shorter cycles of the consumer

real wage, it can be noted that for shorter cycles the producer real wage seems to lead

22

Figure 5: Phase angle: real GDP and producer real wage cycles

0.0 0.2 0.4 0.6 0.8 1.0

HP cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

BK cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

BN cyclesπ

−π

π/2

−π/2

0

0.0 0.2 0.4 0.6 0.8 1.0

STSM cyclesπ

−π

π/2

−π/2

0

Notes: The horizontal axis represents (angular) frequency ω.

the corresponding real GDP cycle. The main problem, however, is the insignificance

of almost all estimates. Hence, the results for producer real wages in the frequency

domain remain inconclusive.

5 Summary and Conclusions

This paper provides stylized facts about the cyclical behavior of consumer and pro-

ducer real wages in Germany. In order to see whether a robust empirical picture

on real wage behavior emerges, several detrending methods have been applied to

23

both real wage series and real GDP, including the deterministic trend model, the

Beveridge–Nelson decomposition, the Hodrick–Prescott filter, the Baxter–King filter

and the structural time series model. The stochastic properties of the original time

series and the derived cyclical components were analyzed using a set of unit root

tests and other diagnostic tests. Since the cycles generated by the deterministic

trend model violated the stationarity condition, they were excluded from further

analysis.

We then analyzed the comovements of the detrended real wage series with real

GDP in the time domain and in the frequency domain. For both approaches not

only the contemporaneous correlation between real wages and GDP, but also the

lag–lead structure has been taken into account. In the time domain the sample cross–

correlations between the cycle of each of the real wage series and the GDP cycle have

been evaluated. According to our results in the time domain, the contemporaneous

correlation between the real wage and GDP is statistically insignificant, with the

exception of the cycles from the structural time series model. In the latter case we

found a negative contemporaneous correlation. Regarding the lead–lag structure, the

consumer real wage displays a procyclical pattern and lags behind the business cycle.

The strongest reaction to the actual economic situation can be observed between the

5th and the 11th quarter. For the producer real wage all estimated contemporaneous

cross–correlations are statistically insignificant. Furthermore, although there is a

similar cyclical pattern as in the case of the consumer real wage, the sample cross–

correlations at the leads of the real wage are not as high.

In the next step, we analyzed the comovements in the frequency domain. The

great advantage of an analysis in the frequency domain is that it allows to assess

the relative importance of particular frequencies for the behavior of real wages. We

followed the non-parametric approach to the estimation of spectra and cross–spectra.

The analysis of the phase angle for the consumer real wage shows that the observed

24

cyclicality depends on the frequency range under consideration. All decomposition

methods for which we got statistically significant results reveal a similar pattern.

The consumer real wage is lagging the real GDP cycle. For shorter time periods

up to about three years, the consumer real wage shows an anticyclical behavior,

whereas for longer time spans a procyclical behavior can be observed. However, for

the producer real wage the results in the frequency domain remain inconclusive.

Our results for consumer real wages are in line with an economy that is char-

acterized by wage stickiness in the short run. For example, an economic upswing

could first lead to a decline in real wages because of rising prices and rigid nominal

wages. In the longer run, nominal wages are adjusted upwards eventually leading

to a rise in real wages as well.

25

A Data Selection

We use quarterly data for Germany from 1970.Q1 to 2009.Q1 (157 observations).

All series that served to generate the project data, except for working hours, have

already been available as seasonally adjusted data based on the Census-X12-ARIMA

procedure.

Real GDP

In order to obtain the real GDP series we used the price adjusted chain index with

the base year 2000 (source: Deutsche Bundesbank, series JB5000). The raw data

before 1991.Q1 referred to West Germany and after 1991.Q1 to unified Germany.

The index series has already been linked over the annual average for 1991. We

multiplied the index with the nominal GDP in 2000 and divided it by 100 (source

of nominal GDP: Statistisches Bundesamt, GENESIS online database).

Real wages

We obtained the real wage series on the basis of gross wages and salaries (source:

prior to 1991.Q1 Statistisches Bundesamt, Beiheft zur Fachserie 18, Reihe 3; from

1991.Q1 on Statistisches Bundesamt, GENESIS online database). Since we were

interested in hourly real wages, we divided this series by total working hours of the

domestic labor force. The data for working hours from 1970.Q1 to 1991.Q4 referred

to West Germany (source: Statistisches Bundesamt, Erganzung zur Fachserie 13,

Reihe S.12) and from 1991.Q1 on to unified Germany (source: Statistisches Bun-

desamt, GENESIS online database). After seasonal adjustment with the Census-

X12-ARIMA procedure we linked both series over the annual average for 1991. The

nominal hourly wage has been deflated with the consumer price index (CPI) or the

producer price index (PPI) in order to generate the respective real wage series. The

source of both price indices is Deutsche Bundesbank (CPI: series USFB99, PPI:

series USZH99).

26

B Figures: nonparametric spectral estimates

Figure B.1: Spectra of the real GDP cycles

HP

0.0 0.5 1.0 1.5 2.0

0.5

1.0

1.5

HP BK

0.0 0.5 1.0 1.5 2.0

0.5

1.0

1.5 BK

BN

0.0 0.5 1.0 1.5 2.0

0.5

1.0

1.5 BN STSM

0.0 0.5 1.0 1.5 2.0

1

2

3 STSM

Notes: The horizontal axis represents the (angular) frequency ω.

The values on the vertical axis have been multiplied by 104.

27

Figure B.2: Spectra of the consumer real wage cycles

HP

0.0 0.5 1.0 1.5 2.0

0.25

0.50

0.75

1.00 HP BK

0.0 0.5 1.0 1.5 2.0

0.25

0.50

0.75

1.00 BK

BN

0.0 0.5 1.0 1.5 2.0

1

2

BN STSM

0.0 0.5 1.0 1.5 2.0

1

2

3 STSM

Notes: The horizontal axis represents the (angular) frequency ω.

The values on the vertical axis have been multiplied by 104.

28

Figure B.3: Spectra of the producer real wage cycles

HP

0.0 0.5 1.0 1.5 2.0

1

2

3

HP BK

0.0 0.5 1.0 1.5 2.0

1

2

3 BK

BN

0.0 0.5 1.0 1.5 2.0

1

2

3 BN STSM

0.0 0.5 1.0 1.5 2.0

2

4

6STSM

Notes: The horizontal axis represents the (angular) frequency ω.

The values on the vertical axis have been multiplied by 104.

29

Figure B.4: Cospectra and quadrature spectra between the real GDP cycles

and the consumer real wage cycles

HP

0.0 0.5 1.0 1.5 2.0

0.00

0.25

CospectrumHP HP

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50

0.75 Quadrature spectrumHP

BK

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50BK BK

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50

0.75BK

BN

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5BN BN

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50 BN

STSM

0.0 0.5 1.0 1.5 2.0

−0.50

−0.25

0.00STSM

STSM

0.0 0.5 1.0 1.5 2.0

0.25

0.75 STSM

Notes: The horizontal axis represents the (angular) frequency ω.

The values on the vertical axis have been multiplied by 104.

30

Figure B.5: Cospectra and quadrature spectra between the real GDP cycles

and the producer real wage cycles

HP

0.0 0.5 1.0 1.5 2.0

0.00

0.25

Cospectrum

HP

HP

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50Quadrature spectrum

HP

BK

0.0 0.5 1.0 1.5 2.0

0.00

0.25BK

BK

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50 BK

BN

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5BN BN

0.0 0.5 1.0 1.5 2.0

−0.1

0.1 BN

STSM

0.0 0.5 1.0 1.5 2.0

−0.25

0.00

0.25STSM

STSM

0.0 0.5 1.0 1.5 2.0

0.00

0.25

0.50STSM

Notes: The horizontal axis represents the (angular) frequency ω.

The values on the vertical axis have been multiplied by 104.

31

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